Abstract

Questions of energy dissipation or friction appear immediately when addressing the problem of a body moving in a fluid. For the most simple problems, involving a constant steady propulsive force on the body, a straightforward relation can be established balancing this driving force with a skin friction or form drag, depending on the Reynolds number and body geometry. This elementary relation closes the full dynamical problem and sets, for instance, average cruising velocity or energy cost. In the case of finite-sized and time-deformable bodies though, such as flapping flyers or undulatory swimmers, the comprehension of driving/dissipation interactions is not straightforward. The intrinsic unsteadiness of the flapping and deforming animal bodies complicates the usual application of classical fluid dynamic forces balance. One of the complications is because the shape of the body is indeed changing in time, accelerating and decelerating perpetually, but also because the role of drag (more specifically the role of the local drag) has two different facets, contributing at the same time to global dissipation and to driving forces. This causes situations where a strong drag is not necessarily equivalent to inefficient systems. A lot of living systems are precisely using strong sources of drag to optimize their performance. In addition to revisiting classical results under the light of recent research on these questions, we discuss in this review the crucial role of drag from another point of view that concerns the fluid-structure interaction problem of animal locomotion. We consider, in particular, the dynamic subtleties brought by the quadratic drag that resists transverse motions of a flexible body or appendage performing complex kinematics, such as the phase dynamics of a flexible flapping wing, the propagative nature of the bending wave in undulatory swimmers, or the surprising relevance of drag-based resistive thrust in inertial swimmers.

Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.

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Schematic diagram of the flow…

Figure 1.

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Schematic diagram of the flow streamlines over an airfoil section showing the boundary…

Figure 1.

Schematic diagram of the flow streamlines over an airfoil section showing the boundary layer and its separation on one side defining the width of the near wake. The drag force is in the direction of the uniform flow velocity U far from the streamlined object, lift is perpendicular.

Figure 2.

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Skin friction and pressure drag…

Figure 2.

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Skin friction and pressure drag contributions to the total drag coefficient for a…

Figure 2.

Skin friction and pressure drag contributions to the total drag coefficient for a family of struts of length L and thickness th at Re = 4 × 105 (data from [45]). The CD data were obtained by dividing drag-per-unit-length data by . Drag-per-unit-length divided by thickness is thus here equivalent to drag FD divided by the frontal surface S in equation (1.3). (Online version in colour.)

Figure 3.

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( a ) Trailing stream-wise…

Figure 3.

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( a ) Trailing stream-wise vortices in the wake of a rectangular wing…

Figure 3.

(a) Trailing stream-wise vortices in the wake of a rectangular wing (from [63]). (b,c) Stream-wise vortices detached from model undulatory swimmers of two different aspect ratios—(b) H/L = 0.3 and (c) H/L = 0.7, the foils are shown from behind, i.e. swimming into the plane shown (from [64]). (Online version in colour.)

Figure 4.

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Edge vortices along the body…

Figure 4.

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Edge vortices along the body of a model fish: ( a ) instantaneous…

Figure 4.

Edge vortices along the body of a model fish: (a) instantaneous stream-wise vorticity slices and (b) pressure field over the cross-section indicated as a dashed line in (a) (adapted from [70]). (Online version in colour.)

Figure 5.

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Phase diagrams of drag-driven and…

Figure 5.

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Phase diagrams of drag-driven and added mass-driven propulsion as a function of the…

Figure 5.

Phase diagrams of drag-driven and added mass-driven propulsion as a function of the aspect ratio and slip ratio for (a) anguilliform kinematics and (b) carangiform kinematics. The dashed line represents the 〈Tma〉 = 〈Td〉 in the phase space. Experimental data are obtained from: Gray [72], Tytell [73] and Hess [74] for anguilliform swimmers, and Bainbridge [75], Webb [76], Videler [77] and Videler [78] for caranguiform swimmers. (Adapted from [68].) (Online version in colour.)

Figure 6.

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Schematic diagram of the fluid…

Figure 6.

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Schematic diagram of the fluid and solid dynamics two-way coupling in a fluid–structure…

Figure 6.

Schematic diagram of the fluid and solid dynamics two-way coupling in a fluid–structure interaction problem (adapted from [79]). (Online version in colour.)

Figure 7.

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Beam model for a flapping…

Figure 7.

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Beam model for a flapping wing ( a ) or an undulatory swimmer…

Figure 7.

Beam model for a flapping wing (a) or an undulatory swimmer (b). In (a), the beam represents a section of the wing, shown schematically undergoing a deformation well described by the first mode of a clamped–free beam [82,83]. In (b), the deformation of the beam in a higher mode is represented by the undulatory kinematics of the self-propelled swimmer described in [61,84]. The characteristic velocity of the imposed actuation Aω and the resulting cruising velocity U are represented schematically in both cases. Additionally indicated: for the flapping wing, the angle ϕ that characterizes the ratio of these two velocities; and for the undulatory swimmer, the phase velocity of the bending wave vφ. The length L of the beam is thus the wing chord in (a) or the swimmer body length in (b). (Online version in colour.)

Figure 8.

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Thrust production of ( a…

Figure 8.

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Thrust production of ( a ) heave, ( b ) pitch and (…

Figure 8.

Thrust production of (a) heave, (b) pitch and (c) pitch–heave motions. Comparison between experiments and simulations. Ratio of reactive thrust to total thrust for the simulations in a self-propelled configuration as a function of the non-dimensional flapping frequency and the plate aspect ratio: (d) heave (e) pitch and (f) pitch–heave. The definition of T+am here is slightly different from 〈Tma〉 in figure 5 because it includes only positive contributions of the local force to the integral (see [93] for details; adapted from [93]). (Online version in colour.)

Figure 9.

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( a , b )…

Figure 9.

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( a , b ) Vibration experiments described in [94] performed on a…

Figure 9.

(a,b) Vibration experiments described in [94] performed on a Mylar plate flapped with a shaker in air (a) and in water (b). In air, a standard standing-wave solution is observed, that is characteristic of systems influenced by the boundary conditions. In water, with a stronger damping, the plate now exhibits a travelling solution. (c–h) Dynamics of the elastic undulatory swimmer described in [61]. (c) Definition of the coordinates and geometry of the beam model. (d–f) Simulated motion of the beam when implementing equation (4.2) gradually: (d) with only the two first terms describing a classic elastic beam (e) adding the ‘flag’ terms in brackets, (f) adding the quadratic fluid term. (g) Successive computed shapes superimposed to pictures of a 4.5 cm long swimmer forced at f = 19 Hz. (h) Experimental envelope to be compared with the computed envelope in (f). Scale bar in (a,b) is 1 cm. (Adapted from [61,94].) (Online version in colour.)

Figure 10.

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( a ) Photograph of…

Figure 10.

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( a ) Photograph of a flapping wing from Ramananarivo et al. [83]…

Figure 10.

(a) Photograph of a flapping wing from Ramananarivo et al. [83] showing successive states of the bending wing during one stroke cycle. As can be seen, the deformation is mainly performed on the first mode. In this case, the phase lag is quite large, leading to a strong increase of flight performance. Scale bar is 1 cm. (b–c) Evolution of the non-dimensional amplitude (b) and phase (c) of the trailing edge wing response as a function of the reduced driving frequency for two flapping amplitudes A = 0.8L and A = 0.5L (filled symbols correspond to measurements in air, open symbols in vacuum). Those results are compared to nonlinear predictions from equation (4.3) with (grey line) and without (black line) nonlinear air drag. The vertical grey band in (b) and (c) marks the optimum of performance quantified by a dimensionless thrust power (see [83] for details; adapted from [83]). (Online version in colour.)